C₂×D₃² #649

order	72
isoclinism	isoclinic class of D₃²
automorphism	Aut(G) = #64241 (order 288) · Inn(G) = D₃² · Out(G) = D₄
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

direct product	C₂×D₃²
unambiguous semidirect product	(C₃×D₃)⋊C₂²
unambiguous semidirect product	(C₃⋊D₃)⋊C₂²

Fitting, socle	C₂×C₃² ⋊ C₂²
derived, Sylow	C₃² ⋊ C₂³

	subgroup	G/H
center, hypercenter	C₂	D₃²
commutator, cosocle, hypocenter	C₃²	C₂³
Frattini	C₁	G
Fitting, socle	C₂×C₃²	C₂²
LCS	G ▷ C₃²
UCS	C₁ ◁ C₂

69 conjugacy classes of subgroups.

minimal degree	μ(G) = 8
transitive degree	μ_t(G) = 12

A minimal faithful permutation representation — 3 generators on 8 points:

(1,2)(4,5)(6,7,8);(3,4,5)(7,8);(1,2)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(72, 46);